This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This paper presents an acceleration slip regulation (ASR) system for four-wheel drive (4WD) electric vehicles, which are driven by the front and rear axles simultaneously. The ASR control strategy includes three control modes: average distribution of inter-axle torque, optimal distribution of inter-axle torque and independent control of optimal slip rate, respectively, which are designed based on the torque adaptive principle of inter-axle differential and sliding mode control theory. Furthermore, in order to accurately describe the longitudinal tyre force characteristic, a slip rate calculation formula in the form of a state equation was used for solving the numerical problem posed by the traditional way. A simulation was carried out with the MATLAB/Simulink software. The simulation results show that the proposed ASR system can fully use the road friction condition, inhibit the drive-wheels from slipping, and improve the vehicle longitudinal driving stability.

Although the dynamic performance of four-wheel drive (4WD) conventional vehicles is improved compared with that of two-wheel drive vehicles, the fuel economy is relatively poor. It’s worth noting that a purely electric vehicle can completely ignore the impact of fuel economy [

Electric vehicles have several advantages such as a drive wheel torque that can be acquired easily, sensitive motor response, and motor torque that can be controlled accurately,

In this paper, the 4WD electric vehicle ASR system layout was as shown in

The four-wheel drive (4WD) electric vehicle acceleration slip regulation (ASR) system layout.

Since this paper focuses on the influence of ASR on the vehicle’s longitudinal dynamic performance, the vehicle dynamics model adopted here contains five degrees of freedom as seen in

(1) Vehicle longitudinal movement:

where _{xi}

(2) Four wheel’s rotation movement:

where ω_{wi}_{wi}_{w}_{mi}

The tyre dynamics are usually modeled with the “Magic Formula” developed by Pacejka _{xi}

where _{z}_{i}

“Magic Formula” fitting coefficients.

No. | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

_{i} |
1.02316 | 21.80968 | 526.2336 | 0.09624 | 250.33146 | 0.00906 |

– | ||||||

_{i} |
−0.00255 | 0.03726 | 0.87693 | −0.00009 | −0.00033 | – |

In this paper, permanent magnet synchronous motors were chosen for constructing the dual-motor drive system. The motor’s rated torque is 270 Nm and the rated power is 90 kW. Motor models are usually mainly divided into the theoretical model and the quasi- steady-state model. This paper adopted the latter one and plotted the motor external characteristic curve according to the characteristic motor parameters, and acquired the motor speed and peak torque according to the load signal namely the accelerator open degree, then adopted one-order inertia to complete the dynamic correction on the motor output torque as the following equation:

where _{m}_{max} is the motor’s peak torque, α is the degree of openness of the accelerator, τ_{m}_{N}

The tyre slip rate is the key factor for tyre longitudinal force calculation, the relationship among slip rate, wheel speed and vehicle velocity can be expressed as follows:

The wheel speed ω_{wi}_{i}_{wi}

where σ_{x}_{s}_{x}_{s}_{x}

The simulation results of tyre slip rate are shown in

As shown in _{r}_{i}_{x}

where ∆λ_{i}_{i}_{low} is the low vehicle velocity threshold.

Comparison of two methods for tyre slip rate calculation: (

The simulation result of the modified tyre slip rate is shown in

The modified tyre slip rate at a low vehicle velocity.

The cause of tyre slip phenomena is that the wheel drive torque exceeds the torque which is provided by the road adhesion limitation, and its essence is the balanced relationships between maximum drive torque, road adhesion conditions and the driver-desired torque. For the 4WD electric vehicle in this paper, the maximum drive torque is the motor’s maximum torque, the road adhesion conditions are expressed by the product of the road drive force and wheel radius, namely _{road} = _{x}_{w}_{d}_{pedal} (0–1), namely _{desire} = α_{pedal} × (_{f_}_{max} _{r_}_{max}), where _{f_}_{max} and _{r_}_{max} are the maximum torque values of the front and rear motors, and both of them are equal due to their identical characteristic parameters, namely _{f_}_{max} = _{r_}_{max} = _{max}. So it can be simplified that _{d}_{pedal} ×2 × _{max} and evidently _{d}_{max}. The specific analysis of the balance relationships between maximum drive torque, road adhesion conditions and driver-desired torque is given as follows:

_{d}_{max} ≤ _{road}, generally corresponds to a condition of high road adhesion. In this case, the drive wheel slip will not happen over the entire range of the accelerator pedal, and it’s not necessary to perform a complex torque distribution for the front and rear axles, just let α_{f}_{r}_{pedal}. Although the slip rates of the front and rear axles will be different with the same drive torque due to the different vertical loads of front and rear axles, the vehicle dynamic performance and driving stability will not be influenced.

_{d}_{road} ≤ 2 × _{max}, generally corresponds to the condition of middle road adhesion. In this case, although the driver-desired torque _{d}_{road}, the slip phenomenon may happen on the driving shaft with a smaller axle load due to the bad road conditions. So a reasonable allocation of the drive torque of the front and rear axles is required to ensure _{f}_{r}_{d}

2 × _{max} ≥ _{d}_{road}, generally corresponds to the condition of low road adhesion. In this case, the driver-desired torque _{d}_{road}, and the inter-axle torque distribution will cause an unavoidable slip phenomenon. In order to make full use of the road adhesion, the independent control for front and rear axles is the best control plan, thus the slip rates of the front and rear axles can be controlled to be equal to the optimal rate slip. In this way _{f}_{r}_{road}, and the vehicle can acquire the biggest power at this moment.

In a word, the control flow of acceleration slip regulation strategy is as shown in

The acceleration slip regulation strategy.

The specific control algorithm is mainly divided into three control modes as follows.

This control mode equally assigns the demand torque into the front and rear axle:
_{fa}_{ra}_{desire} / 2

where _{fa}_{ra}

If the slip phenomenon occurred, this control mode will be adopted. The slip phenomenon determining condition is as follows: the front axle slip rate λ_{f}_{r}_{o}_{o}_{o}_{o}_{o}

where _{o}_{o}_{f}_{r}

It can be found by substituting Equations (2) and (12) into Equation (11) after derivation:

where _{fo}_{ro}

If the slip phenomenon still occurred after Mode 2 control was adopted, this control mode will be adopted here to control the two motors independently and ensure the slip rate of front and rear axle are equal. The switching functions _{fi}_{ri}_{fi}_{ri}

where λ_{o}_{fi}_{ri}_{fi}_{ri}

It can be found using the same reasoning that:

Under the premise of ensuring the vehicle’s longitudinal stability, acceleration slip regulation is proposed to improve the vehicle’s dynamic performance. To verify the validity and accuracy of the control strategy, first, different control modes for fixed conditions were subjected to separate simulation analyses, then the effective switching between different control modes for variable condition were verified by simulation. The simulation experiments were carried out using the MATLAB/Simulink (MathWorks, Natick, MA, USA) environment, and the model parameters are as listed in

Simulation parameters.

Parameters | Values | Parameters | Values |
---|---|---|---|

_{o} |
500 | _{ri} |
80 |

_{fi} |
50 | 5000 kg | |

_{ri} |
500 | _{N} |
4000 |

_{s} |
1.81 kg·m^{2} |
_{w} |
0.447 m |

_{x} |
1.65 kg·m^{2} |
_{low} |
1.83 m/s |

_{w} |
2.035 kg·m^{2} |
τ_{m} |
200 |

_{o} |
−100 | σ_{x} |
0.91 |

_{fi} |
−200 | – |

When the vehicle is traveling on a high adhesion road, the average distribution of inter-axle torque model is used to control the load signal of the front and rear axle drive motor α_{f}_{r}_{pedal}. The simulation results are shown in _{pedal} = 0.5 and the peak road adhesion coefficient μ_{H} = 0.80.

As ^{2} under these conditions, then due to the motor speed increase, the vehicle acceleration will be reduced gradually when the motor workspace transforms from the constant torque to the constant power region.

Although the slip rate can be guaranteed to be a constant based on the torque distribution by the front and rear axle loads, in the view of the project realization, the accurate estimation of dynamic load of the front and rear axle when the vehicle is running requires a large number of external sensors, so the controller cost is expensive. In contrast, the average distribution of inter-axle torque control is simple and low cost,

Simulation results of average distribution of inter-axle torque on good roads: (

Simulation results of torque distribution by axle load on good roads: (

When the driver steps on the acceleration pedal lightly on a mid-adhesion road, although the driver demanded torque is less than the total road adhesion at that time, if the average distribution of inter-axle torque control is still used, the drive axle which has the smaller axle load suffers a smaller ground tangential force, and the corresponding road adhesion is smaller, which may seriously slip due to its relatively large driving torque. Similarly, the drive axle which has smaller axle load has smaller road adhesion, and the corresponding drive torque is smaller, resulting in it being unable to take full advantage of the road adhesion conditions. Therefore, the design is based on an optimal distribution of inter-axle torque to have a reasonable distribution of the drive torque between the front and rear axles. The simulation results of slip rate between front and rear axle under the optimal distribution of inter-axle torque control and without torque control are shown in _{pedal} = 0.30 and peak road friction coefficient μ_{H} = 0.10.

^{2} under the optimal distribution of inter-axle torque control based on SMC (Slip Mode Control), while the vehicle acceleration eventually stabilizes at 0.65 m/s^{2} under the average torque distribution control. In poor road conditions, the optimal distribution of inter-axle torque control based on SMC was significantly better than the average torque distribution control, not only ensuring the vehicle’s longitudinal driving stability, but also improve the vehicle’s dynamic performance.

Simulation of stepping on the pedal lightly to accelerate on a mid-adhesion road: (

The sliding mode control process under these conditions is shown in

The change of sliding mode surface and controlled variables in the sliding mode control process: (

When the vehicle drives on s bad road and the driver-demanded torque is large, _{desire} > _{road}, tyre slip can’t be avoided with the torque distribution between the axis based on optimal distribution of inter-axle torque control, so independent control of the optimal slip rate is required, the front and rear axle slip rate remains the best slip rate (referred to herein λ_{opt} = 15%), in order to fully use the road adhesion conditions and get maximum power. The signal of the driver accelerator pedal is set at α_{pedal} = 0.8, and the peak road adhesion coefficient is set at μ_{H} = 0.10, and the independent control of optimal slip rate and the optimal distribution of inter-axle torque control results are shown in

Simulation results of stepping on the pedal heavily on snowy roads: (

^{2} in this mode, as shown in _{opt} = 15%), but also maximize use of the road adhesion, to obtain an acceleration of 0.98 m/s^{2} which was shown in

The front and rear axle independent control process under this condition is shown in

Front and rear axle input torque under different control modes: (

To further verify that the ASR control strategy can be effective for three different conditions to switch under constant driver-demanded torque, changing road conditions, and driver demanded torque changes, the road conditions remain unchanged to simulate two conditions. The simulation results are shown in

_{pedal} = 0.3) by a dry road (μ_{H} = 0.80) to start with icy ground (μ_{H} = 0.10), at about 3.8 s the controller detects front axle tyre slip λ > 15%, so it switches the average distribution mode to the axis torque distribution based on SMC mode, as shown in _{desire} < _{road} is always correct, before the control mode switching, vehicle acceleration always follows the driver’s intention, steady at around 0.78 m/s^{2}, as shown in

Simulation results of torque distribution on changing roads: (

_{pedal} = 0.8) by a dry road ground (μ_{H} = 0.80) starting with icy ground (μ_{H} = 0.10), the controller detects that the front axle slips at about 2.1 s, mode switches from the average distribution of the torque to distribution of inter-axle torque, but _{desire} > _{road} at this time, so distribution of inter-axle torque control is unable to prevent tyre slippage, so it continues to switch from distribution of inter-axle torque mode to the front and rear axles independent control mode, as shown in _{desire} > _{road} at this time, the front and rear axle independent control mode can take advantage of the road adhesion to the optimum dynamic, the acceleration eventually stabilizes at around 0.98 m/s^{2}, as shown in

To further verify the torque distribution strategy between the axis effective switching between the three control modes, we simulated under the same road conditions (μ_{H} = 0.10), the changing driver- demanded torque conditions, as shown in

_{pedal} = 0.1) with pedal signal α_{pedal} = 0.1, the driver-demanded torque is small, the front and rear drive axles are both under road adhesion conditions, and the controller controls the front and rear axle drive motors under average torque distribution mode. Then at about 4 s, the pedal changes to α_{pedal} = 0.3, then the slip rate of the front axle has a slip trend, and then the controller switches to the axis torque distribution based on SMC mode. When the pedal depth further increases until α_{pedal} = 0.8, the total ground adhesion is not enough, so slippage can’t be avoided by torque distribution between the axles, so the controller switches to the front and rear axles independent control mode. With increasing speed, the motor transforms from constant torque region to constant power region, under the constant accelerator pedal, the driver-demanded torque is decreasing, then the mode switches to “the axis torque distribution based on SMC mode” again, as shown in

It can thus be seen that the proposed inter-axis torque distribution control strategy can effectively switch between different conditions, ensuring the vehicle dynamics while improving the vehicle's longitudinal driving stability.

Simulation results under the same road with the changing driver accelerator pedal: (

Aiming at the 4WD electric vehicle, which was driven by front and rear independent motors, a model of the ASR system was established.

Compared with the conventional method of slip rate calculation, using the state equation of slip rate can be more accurate to describe the tyre slip process in a low vehicle velocity situation.

An ASR control strategy which contains three torque distribution mode was designed, namely average distribution of inter-axle torque for high road adhesion, optimal distribution of inter-axle torque for middle road adhesion and independent control of optimal slip rate for low road adhesion. Several simulations were carried out with MATLAB/Simulink, and the simulation results with some comparisons show that, the proposed strategy could realize the transformation among different control modes, thus fully use the road adhesion conditions, make the vehicle’s dynamic performance to follow the driver’s wishes. As a result, the vehicle longitudinal drive stability and dynamic performance are ensured.

This work was supported by the National High Technology Research and Development Program of China (2012AA111603, 2013BAG05B00) in part, the Program for New Century Excellent Talents in University (NCET-11-0785) and Beijing Institute of Technology Post Graduate Students Innovation Foundation in part. The author would also like to thank the reviewers for their corrections and helpful suggestions.

Hongwen He and Jiankun Peng built the dynamic model of ASR for 4WDEV and performed the simulations, Rui Xiong designed the control modes, Hao Fan debugged the algorithm. All authors carried out data analysis, discussed the results and contributed to writing the paper.

The authors declare no conflict of interest.